Formal power series

DEFINITION 01.3   Let $A$ be a commutative ring. Let $X$ be a variable. A formal power series in $X$ over $A$ is a formal sum

$$\sum_{i=0}^\infty a_i X^i \quad ( a_i \in A )$$

We denote by $A[[X]]$ the ring of formal power series in $X$ . Namely,

$$A[[X]]=\{ \sum_{i=0}^\infty a_i X^i; a_i \in A\}.$$

For any element $f=\sum_n a_n X^n$ of $A[[X]]$ , we define its order as follows:

$${\mathrm{ord}}(f)=\inf\{n; a_n \neq 0\}.$$

Then we may define a metric on $A[[X]]$ .

$$d(f,g)=\frac{1}{2^{{\mathrm{ord}}(f-g)}}$$

EXERCISE 01.1   Show that $(A[[X]],d)$ is a complete metric space.

EXERCISE 01.2   Show that $A[[X]]$ is a topological ring. That means, it is a topological space equipped with a ring structure and operations (the addition and the multiplication) is continuous.